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Kusumah, Ferdi Perdana; Kyyrä, JormaMinimizing coil power loss in a direct AC/AC converter-based contactless electric vehiclecharger

Published in:Proceedings of the 19th European Conference on Power Electronics and Applications, EPE'17 ECCE Europe

DOI:10.23919/EPE17ECCEEurope.2017.8099168

Published: 09/11/2017

Document VersionPeer reviewed version

Please cite the original version:Kusumah, F. P., & Kyyrä, J. (2017). Minimizing coil power loss in a direct AC/AC converter-based contactlesselectric vehicle charger. In Proceedings of the 19th European Conference on Power Electronics andApplications, EPE'17 ECCE Europe (pp. 1-10). (European Conference on Power Electronics and Applications).IEEE. https://doi.org/10.23919/EPE17ECCEEurope.2017.8099168

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Minimizing Coil Power Loss in a Direct AC/AC Converter-basedContactless Electric Vehicle Charger

Ferdi Perdana Kusumah and Jorma KyyraDepartment of Electrical Engineering and Automation

School of Electrical EngineeringAALTO UNIVERSITY

P.O. Box 13000FI-00076 Aalto, Finland

Email: ferdi.kusumah@aalto.fiURL: http://eea.aalto.fi/en/

KeywordsAC/AC converter,Battery charger,Contactless power supply,Electric vehicle,Resonant converter,Soft switching.

AbstractThis paper explains an optimum load analysis to reduce power loss in a direct AC/AC converter-basedcontactless electric vehicle charger. The converter has a fewer number of bi-directional switches than amatrix converter and it uses a resonant circuit to utilize zero-current switching. Output power and powerloss depend on coupling factor and output load. A proper load value leads to low loss and maximumtransmission efficiency. The value was derived using an iterative-based Weierstrass’ method due toconverter dynamic switching frequency. Other parameters such as link efficiency, link gain and powerloss ratio were derived based on steady-state analysis. Simulation results are then presented to validatethe theoretical analyses.

IntroductionInductive-based contactless power transfer (ICPT) has been developed globally to charge an electricvehicle (EV). The main purpose behind its usage is to remove direct wire connections between thecharger and the vehicle which leads to reliability, mobility, and safety improvements. Deployment of thecharging system can improve EV’s charging experience which may lead to an increased popularity andadoption of the vehicle [1]-[3].

In a typical series-series compensated ICPT system connected to a three-phase source, a rectifier and aDC/AC converter are used in a primary side to drive primary coil current. To minimize power loss inprimary and secondary coil resistances, a proper optimum load value must be selected. An additionalconverter at the input side is also added to reduce primary current at a partial load that cannot be achievedby either frequency or dual control method [3]-[5]. This solution adds extra cost to the charging system.

In [6] and [7], the authors proposed a direct AC/AC converter for ICPT that has a lesser number ofswitches than a matrix converter. It operates based on current injection and free-oscillation strategieswhich were previously introduced in [8]-[10]. Since the output power is controlled by a primary cur-rent amplitude, the system can reduce coil power loss without an additional input-side converter. Theswitching frequency is based on system resonant frequency to utilize zero-current switching (ZCS), sothe value depends on a coils coupling factor as well as a connected load. Due to variable frequency, theoptimum load calculation is more complicated than in a series-series compensated ICPT case given in[5]. In addition to the load analysis, link efficiency, link gain and power loss ratio equations will also be

investigated. All of them will be used to formulate current reference to control the output power at lowcoil power loss.

This paper is organized as follows. A brief description of the proposed direct AC/AC converter topologyand its working principle are given in Section Contactless charger system. Theoretical formulationsgive explanation on how to calculate the optimum load using steady-state analysis approach to achievelow power loss. In Section Simulation results, PLECS software outcomes are presented to verify thetheoretical calculations. Finally, conclusions of the paper are given in the last section.

Contactless charger systemConverter topology and its equivalent circuit are given in Fig. 1. In topology figure, the primary sidecontains four bi-directional switches (Sa, Sb, Sc and Sd). The switches will be used to drive the resonantcircuit Lp, Cp and Ls which are controlled by the on-off current controller. The controller receives sensorinformation marked by dashed lines, which are three-phase input voltage vx(t), primary side resonantcurrent ip(t), and connected battery voltage vo(t). It will change the amplitude of ip(t) based on batteryvoltage level provided by the secondary pick-up circuit. The topology can be simplified for modelingpurpose by an equivalent circuit model given in Fig. 1b. Parameters Rp and Rs indicate primary andsecondary coil resistances respectively. For a particular type of modulation strategy explained in [6], theinput side can be approximated by a square wave vin(t). A load Req is a representation of the secondarycircuit rectifier and battery obtained from [11], where the battery is assumed to be purely resistive (RL).

Sw4Sw2

Sw3

Sw1

LsLp

va

vb

vc

On-Off CurrentController

SecondaryPick-upCircuit

Va

Vb

Vc

Lp

C

VoltageSensors

CurrentSensor

Va

Vb

Vc

Lp

C

START

Is I_out > I_ref ?

Max(|Va|, |Vb|, |Vc|) Turn on Sw4 (Free-oscillation)

No Yes

PWM to Sw1/2/3

Va/b/c

Sw1

Sw2

Sw3

Sw4

I_out

I_out

AC source Rectifier Inverter RectifierElectric vehiclebattery

Cs

LsLp

Va

Vb

Vc

Cp

On-Off CurrentController

VoltageSensors

CurrentSensor

Vo

Sw1

Sw3

Sw2

Sw4

vo

Secondary circuit sidePrimary circuit side

vx ip

sx

vo

Sa

Sc

SbSd

AC source

Rectifier Inverter RectifierElectric vehiclebatteryCDC

Secondary circuit sidePrimary circuit side

Cp

(a) Topology

CpReqp

Vin Cs RLLp Ls

ip is

CpReqp

Vin ReqsLp Ls

ip is

M

M

Cp Cs RLLp Ls

ip is

ReqLs

M

Vs Φ

Cp Lp

M

Vs ΦCp

Lp

ip

Vs Φ

RrefCref

ip is

ReqLsCp Lp

M

vs

ip is

Req = RL/2

LsCp Lp

M

vin

ip is

RsRp

Req = 8RL/π2

Ls

Cp

Lp

M

vin(t)

ip(t)

is(t)

RsRp

Req = 8RL/π2

Ls

Cp

Lp

M

ip(t)

is(t)

RsRp

vin(t)

(b) Equivalent circuit

Fig. 1: A Direct three-phase to single-phase AC/AC converter topology and its equivalent circuit.

Some possible current commutations of the primary side are given in Fig. 2. Case 1 and 2 correspondto injection mode where the primary current ip(t) flows either from or to one of the input sources whichis phase “a” in this case. It can be seen that only either Da2, Sa+ or Da1, Sa− pair which isconnected to the corresponding phase is operational. The mechanism of injection modes for other inputphases are also similar. Free-oscillation mode is described in Case 3 and 4, where the current oscillatesin the resonant circuit. The switching transitions between injection and free-oscillation are done duringzero-crossing of the primary resonant current. Both injection and free-oscillation modes will be used tocontrol the current amplitude which leads to a controlled output power. Converter behavior using oneparticular modulation strategy is explained in more detail in [6] and [7]. It basically injects the resonantcircuit from the input that equals “maximum absolute value of all three-phase inputs”. Every injection ismostly accompanied by free-oscillation. Analyses in this paper are based on such modulation strategy.

Theoretical formulationsTo simplify theoretical calculations, all electric components are assumed to be ideal and the load is consi-dered to be purely resistive. The converter is also assumed to be operated in an open-loop configuration.The used modulation strategy produces zero phase-shift between input square voltage and primary reso-

va

vb

vc

Lp

Cp

+ip va

vb

vc

Lp

Cp

ip

va

vb

vc

Lp

Cp

+ip

va

vb

vc

Lp

Cp

ip

Da1 Da2

Db1 Db2

Dc1 Dc2

Sa+

Sb–

Sc–

Sa–

Sb+

Sc+

Sd–

Sd+

Dd2

Dd1

Sa+

Sb–

Sc–

Sa–

Sb+

Sc+

Sd–

Sd+

Dd2

Dd1

Da1 Da2

Db1 Db2

Dc1 Dc2

Sa+

Sb–

Sc–

Sa–

Sb+

Sc+

Sd–

Sd+

Dd2

Dd1

Da1 Da2

Db1 Db2

Dc1 Dc2

Sa+

Sb–

Sc–

Sa–

Sb+

Sc+

Sd–

Sd+

Dd2

Dd1

Da1 Da2

Db1 Db2

Dc1 Dc2

(a) Case 1

va

vb

vc

Lp

Cp

+ip va

vb

vc

Lp

Cp

ip

va

vb

vc

Lp

Cp

+ip

va

vb

vc

Lp

Cp

ip

Da1 Da2

Db1 Db2

Dc1 Dc2

Sa+

Sb–

Sc–

Sa–

Sb+

Sc+

Sd–

Sd+

Dd2

Dd1

Sa+

Sb–

Sc–

Sa–

Sb+

Sc+

Sd–

Sd+

Dd2

Dd1

Da1 Da2

Db1 Db2

Dc1 Dc2

Sa+

Sb–

Sc–

Sa–

Sb+

Sc+

Sd–

Sd+

Dd2

Dd1

Da1 Da2

Db1 Db2

Dc1 Dc2

Sa+

Sb–

Sc–

Sa–

Sb+

Sc+

Sd–

Sd+

Dd2

Dd1

Da1 Da2

Db1 Db2

Dc1 Dc2

(b) Case 2

va

vb

vc

Lp

Cp

+ip va

vb

vc

Lp

Cp

ip

va

vb

vc

Lp

Cp

+ip

va

vb

vc

Lp

Cp

ip

Da1 Da2

Db1 Db2

Dc1 Dc2

Sa+

Sb–

Sc–

Sa–

Sb+

Sc+

Sd–

Sd+

Dd2

Dd1

Sa+

Sb–

Sc–

Sa–

Sb+

Sc+

Sd–

Sd+

Dd2

Dd1

Da1 Da2

Db1 Db2

Dc1 Dc2

Sa+

Sb–

Sc–

Sa–

Sb+

Sc+

Sd–

Sd+

Dd2

Dd1

Da1 Da2

Db1 Db2

Dc1 Dc2

Sa+

Sb–

Sc–

Sa–

Sb+

Sc+

Sd–

Sd+

Dd2

Dd1

Da1 Da2

Db1 Db2

Dc1 Dc2

(c) Case 3

va

vb

vc

Lp

Cp

+ip va

vb

vc

Lp

Cp

ip

va

vb

vc

Lp

Cp

+ip

va

vb

vc

Lp

Cp

ip

Da1 Da2

Db1 Db2

Dc1 Dc2

Sa+

Sb–

Sc–

Sa–

Sb+

Sc+

Sd–

Sd+

Dd2

Dd1

Sa+

Sb–

Sc–

Sa–

Sb+

Sc+

Sd–

Sd+

Dd2

Dd1

Da1 Da2

Db1 Db2

Dc1 Dc2

Sa+

Sb–

Sc–

Sa–

Sb+

Sc+

Sd–

Sd+

Dd2

Dd1

Da1 Da2

Db1 Db2

Dc1 Dc2

Sa+

Sb–

Sc–

Sa–

Sb+

Sc+

Sd–

Sd+

Dd2

Dd1

Da1 Da2

Db1 Db2

Dc1 Dc2

(d) Case 4

Fig. 2: Possible current commutations of the AC/AC converter.

nant current [6]. Circuit model in Fig. 1b is used for the calculations. All power equations refer to anaverage power.

Link efficiency

Primary and secondary circuits transmission/link efficiency analysis is used to make the converter ope-rates in low coil power loss region. By looking at Fig. 1b, the link efficiency, or ηlink is defined as amultiplication between “a ratio of a power transfered to the secondary side to a power put across theprimary circuit” and “a ratio of a power dissipated in the load Req to a total power consumed by thesecondary side” [5]. The parameters can be obtained using a steady-state analysis under open loop confi-guration given in [3] and [5]. Resonant frequency of the system is calculated beforehand. For ICPT thatcontains only a series capacitor on the transmitter side, the steady-state equations are,

Vs =ip

jωCp+ ipjωLp + ipRp− isjωM

0 = isReq + isjωLs + isRs− ipjωM,(1)

where Vs is the root-mean-square (RMS) of vin(t). Their amplitude relationship for a particular modula-tion method used in this paper is explained in [6], which is Vs = VA/π. Variable VA is an amplitude ofaveraged vin(t) during maximum absolute value of three-phase input. The combinations of both equations

in (1) by eliminating either is or ip are described in (2) and (3) as,

Vs = ip

[Rp +

ω2M2(Req +Rs)

ω2L2s +(Req +Rs)2 + j

(ωLp−

1ωCp− ω3M2Ls

ω2L2s +(Req +Rs)2

)], (2)

= is

[(Rs +Req + jωLs

jωM

)(Rp +

1jωCp

+ jωLp

)− jωM

]. (3)

During resonant (ω = ω0), the imaginary part in (2) equals zero, and its frequency can be found bysolving the part,

ω0Lp−1

ω0Cp=

ω30M2Ls

ω20L2

s +(Req +Rs)2 , (4)

Cp(LpL2s −M2Ls)ω

40 +[LpCp(Req +Rs)

2−L2s ]ω

20− (Req +Rs)

2 = 0. (5)

The roots of biquadratic equation can be solved by substitution and produces four resonant frequencies,

ω0 =±√−A±Ω, A =

1L2

p

[m2(Req +Rs)

2−n1− k2

], Ω =

√A2 +

4nm2(Req +Rs)2

L4p(1− k2)

, (6)

m =Lp

Ls, n =

Lp

Cp. (7)

The practical ω0 is one that has positive and real value. In a real case, Req > 0, Rs > 0 and 0 < k < 1,therefore Ω > A is always valid. The practical ω0 equation under such circumstances is,

ω0 =√−A+Ω. (8)

During resonant, primary and secondary current expressions from (2) and (3) become,

ip(res) = Vs/

[Rp +

ω2M2(Req +Rs)

ω2L2s +(Req +Rs)2

], (9)

is(res) = Vs/

RpLs

M+

ω20LsM(Rs +Req)

(Rs +Req)2 +ω20L2

s+ j

[ω3

0L2s M

(Rs +Req)2 +ω20L2

s−

Rp(Rs +Req)−ω20M2

ω0M

].

(10)

The link efficiency is derived during ω = ω0,

ηlink =Preal (p)

Pin

Pout

Preal (s)=

|ip(res)|2Rref

|ip(res)|2(Rp +Rref)

|is(res)|2Req

|is(res)|2(Rs +Req), (11)

=

[ω2

0M2(Req+Rs)

ω20L2

s+(Req+Rs)2

][

Rp +ω2

0M2(Req+Rs)

ω20L2

s+(Req+Rs)2

] Req

(Rs +Req)=

ω20M2Req

ω20M2(Req +Rs)+Rp

[(Req +Rs)2 +ω2

0L2s] , (12)

=1

1+ 1Qsγ

+ γ

k2Qp+ 2

k2QpQs+ 1

k2QpQ2s γ+ 1

k2Qpγ

, (13)

k =M√LpLs

, Qp =ω0Lp

Rp, Qs =

ω0Ls

Rs, γ =

Req

ω0Ls, (14)

where Rref is a real part of reflected secondary side in (2), Preal (p) is a power dissipation over Rref, Pin is aninput power to the primary circuit, Preal (s) is a power dissipation across a combined Rs and Req, and Poutis a consumed power on the secondary load Req. Parameters Qp as well as Qs are primary and secondarysides quality factor, k is a coupling factor, M is a mutual inductance, γ is a load matching factor, and ω0is a resonant frequency of the primary circuit current [3]. The ω0 is not constant in this case but dependson k and Req as given previously in (8).

Parameter Req that corresponds to the maximum link efficiency is represented as Ropt. When Req = Ropt,the power transfer efficiency between primary and secondary circuits is at maximum. It is obtainedthrough utilizing optimum value of γ or γopt. The γopt itself is achieved by equating dηlink

dγwith zero. For

an ICPT with a primary side capacitor, the expression of γopt is,

γopt =

√k2Q2

s +Q2s +1

Qs. (15)

By assuming Qp = Qs and Qp 1, the γopt becomes,

γopt ≈√

k2Q2s +Q2

s

Qs=√

k2 +1. (16)

A combination of equations (7), (8), (16), and the last term in (14) leads to,

Req

ω0Ls=√

k2 +1, (17)

0 =[Ls

√k2 +1

][√−A+Ω

]−Req, (18)

Ψ(Req) = R4eq +Rs(k2 +1)R3

eq +

[R2

s m2−n(k2 +2)2m2

](k2 +1)R2

eq−nRs(k2 +1)2

m2 Req−nR2

s (k2 +1)2

2m2 ,

= 0,

(19)

where its positive and real root is the Ropt. Radicals method of solving quartic equation can be found in[12] and [13]. Since the quartic has many variables, the solution is too complicated to derive.

A different way is to solve all roots simultaneously using Weierstrass’ method, an iterative approach thatis described in [14]. The calculation involves complex number arithmetic. One disadvantage of the met-hod is longer calculation time that can slow down closed-loop controller performance if it is performedon the fly. In practice, the initial values must be selected based on a trade-off between calculation timeand precision. A precalculated lookup table that maps k to Ropt can be an alternative to speed up thecontroller. Root approximation of the quartic equation can be formulated in a following expression,

Req(i) = Req(i)−Ψ(Req(i)

)∏

4j=1, j 6=i

(Req(i)− Req( j)

) , i = 1,2,3,4. (20)

Subscript (i) represents a currently calculated root (four roots in total) while ( j) indicates other rootsexcept (i). Parameters Req and Req constitutes approximative and initial roots respectively. Root approx-imations are obtained by sequentially calculating every root in (19) within a loop. Approximate resultfrom previous calculation is inserted to the next one. Loop continuation is controlled using ∆= Req−Req.Through running the loop and selecting ∆ to a certain precision value, all roots can be obtained simulta-neously [14].

Link gain

Link gain is defined as a ratio of output voltage across secondary load Req to input voltage over primarycircuit [5]. For the ICPT with a series capacitor on the primary under ω = ω0, equation derivations areas follows,

G =

∣∣∣∣∣Vout

Vs

∣∣∣∣∣=∣∣∣∣∣√

PinηlinkReq

Vs

∣∣∣∣∣= ω0MReq

√(Req +Rs)2 +ω2

0L2s

ω20M2(Req +Rs)+Rp

[(Req +Rs)2 +ω2

0L2s] , (21)

=ηlink

√(Req +Rs)2 +ω2

0L2s

ω0M, (22)

=ηlink

k

√√√√ 1m

[(γ+

1Qs

)2

+1

]. (23)

Equation (23) is obtained using descriptions given in equations (7) and (14). The gain at γopt underassumptions that m = 1 and Qp = Qp 1, is given as,

Gopt ≈1k

√k2 +2. (24)

Power loss ratio

Power loss ratio can be defined as a ratio of total power loss in primary and secondary coil resistances tooutput power in the load [3]. In this ICPT case, the ratio is also calculated during ω = ω0. By utilizing(9) and (10), the loss expressions for primary (Pp) and secondary coils (Ps) consecutively are,

Pp = |ip(res)|2Rp, Ps = |is(res)|2Rs. (25)

Power loss ratio is then formulated as follows,

λ =Pp +Ps

Pout. (26)

Analytical calculationsContactless charger parameters to be analyzed is given in Table I. They were obtained from calculati-ons based on limitation analysis given in [7]. Variable Va, Vb and Vc are voltage amplitude from inputphase “a”, “b” and “c” respectively. By looking at the parameters, it can also be determined that variablem and n in (7) have values of 1 and 1000 subsequently. Line frequency of all inputs is given under f . Theparameters will be used to plot link efficiency, gain, output power, and power loss ratio using MATLABsoftware. By utilizing equations (13), (23), (11) and (26), the produced plots are given in Fig. 3 and 4.

Table I: ICPT system component values

Va Vb Vc f Cp Lp Ls Rp Rs k100 V 100 V 100 V 50 Hz 0.2 µF 200 µH 200 µH 0.3 Ω 0.3 Ω 0.55−0.83

From Fig. 3, link efficiency can theoretically be greater than 80% for a certain load range. When the loadgoes to a MΩ scale, the efficiency as well as output power go to zero. In the link gain case, as the loadgets very large, the gain becomes constant. Power loss ratio gets very large as the load becomes muchgreater than 100 Ω. It can be seen that the higher the k value, the higher ηlink becomes, but at the sametime, the output power gets lower.

The value Ropt was acquired by applying Weierstrass’ method using MATLAB. Initial value of each rootand ∆ were set randomly to a complex number of 1+ j. Iteration will stop when ∆≤ 0.1, which was alsochosen randomly. The results in each iteration step for k = 0.55 and k = 0.83 is given in Table II. Forboth cases, Req is the real and positive root, which is 38.698 Ω for k = 0.55 and 47.587 Ω for k = 0.83. Toget Ropt, the values must be converted to RL through RL = Reqπ2/8. Thus for k = 0.55, Ropt is 47.742 Ω,while for k = 0.83, it is 58.708 Ω.

k = 0.55

k = 0.83

k = 0.55

k = 0.83

k = 0.55

k = 0.83

k = 0.55

k = 0.83

Fig. 3: Link efficiency and gain plots with respect to RL using parameters given in Table I. The x-axis islimited to 100 Ω for clarity purpose.

k = 0.55

k = 0.83

k = 0.55

k = 0.83

k = 0.55

k = 0.83

k = 0.55

k = 0.83

Fig. 4: Output power and power loss ratio plots with respect to RL using parameters given in Table I. Thex-axis is limited to 100 Ω for clarity purpose.

For the obtained Ropt values, the link efficiency, link gain, output power and power loss ratio are givenin Table III. The two Gopt were obtained from their corresponding gain plots. It can be seen that thecoupling factor k affects all those parameters. In practice, since Ropt is affected by k, the value mustbe re-calculated if component values are changing due to environmental effect such as temperature orrepositioning of primary and secondary coils.

Simulation resultsPLECS software was used to further validate the theoretical calculations, the results for given componentvalues on Table I are shown in Fig. 5 and 6. The input voltage and currents plots are shown onlyfor k = 0.55 since the waveforms are also the same for k = 0.83, while their amplitudes are different.Parameter va(t), vb(t) and vc(t) indicate the three-phase input voltage while vin(t) marks a voltage overthe primary resonant circuit. From Fig. 5, during positive va(t), Sa+ and Sd+ are operational to produce

Table II: Weierstrass’ method calculation results for two k values

k = 0.55Step Req(1)

Req(2)Req(3)

Req(4)

0 1+ j 1+ j 1+ j 1+ j1 −255.412−32.052j −1.233+3.116j −6.383−0.949j 3.900+0.781j2 −2.421−2.168j 47.643−62.020j −11.188−16.604j 1.825+0.689j3 −0.879−0.640j 30.410+26.814j −26.297+5.972j −0.057+1.378j4 −0.315−0.758j 41.688−6.094j −37.753−0.414j 0.013+0.557j5 −0.310−0.356j 38.655+0.078j −38.746+0.006j −0.098+0.239j6 −0.206−0.176j 38.698−0j −38.750+0j −0.158+0.152j7 −0.171−0.148j 38.698−0j −38.750+0j −0.170+0.149j

k = 0.83Step Req(1)

Req(2)Req(3)

Req(4)

0 1+ j 1+ j 1+ j 1+ j1 −399.385−50.048j −1.178+3.119j −6.308−0.980j 3.563+0.904j2 −2.165−2.307j 67.290−105.718j −9.041−16.067j 1.614+0.790j3 −0.780−0.548j 22.854+30.376j −30.277+20.167j −0.742+1.005j4 −0.151−0.807j 66.733−3.423j −44.740+3.847j −0.355+0.655j5 −0.184−0.485j 47.042−0.662j −47.730−0.048j −0.211+0.299j6 −0.184−0.209j 47.586+0j −47.716+0j −0.192+0.164j7 −0.188−0.148j 47.587+0j −47.717+0j −0.189+0.145j

Table III: Analytical results for two different coupling factors

k = 0.55, Ropt = 47.742 Ω

ηlink Gopt Pout λ

93.696% 2.596 321.859 W 6.728%

k = 0.83, Ropt = 58.708 Ω

ηlink Gopt Pout λ

96.983% 1.924 143.730 W 3.112%

vin(t), while during negative vc(t), only Sc− and Sd− that are being controlled. The same conditions arealso applied for different input phase cases during maximum absolute value condition. Through built-inFourier analysis and mean value calculator in PLECS, some circuit characteristics were extracted andpresented in Table IV. The value of Vs was calculated from fundamental amplitude of vin(t) within6.667 ms time range. Parameter η is a fraction between Pout and Pin.

Table IV: Simulation results for two different coupling factors

k = 0.55, Ropt = 47.742 Ω

Vs 43.840 VrmsIp 8.079 ArmsPin 349.246 WPp 19.583 W

Vout 112.499 VrmsIs 2.907 Arms

Pout 327.069 WPs 2.536 Wη 93.650%

Gopt 2.566λ 6.763%

k = 0.83, Ropt = 58.708 Ω

Vs 43.903 VrmsIp 3.617 ArmsPin 155.548 WPp 3.924 W

Vout 84.619 VrmsIs 1.778 Arms

Pout 150.479 WPs 0.949 Wη 96.741%

Gopt 1.927λ 3.238%

By comparing Table III and IV, it is apparent that both results are similar. The output power depends on

Primary coil power loss (k = 0.55, RL = 47.742 Ohm)

Secondary coil power loss (k = 0.55, RL = 47.742 Ohm)

Output power (k = 0.55, RL = 47.742 Ohm)

Wat

t(W

)

0

20

40

60

Wat

t(W

)

0

2

4

6

× 1e-2Time (s)

0.0 0.5 1.0 1.5

Wat

t(W

)

0

500

1000

Primary coil power loss (k = 0.83, RL = 58.708 Ohm)

Secondary coil power loss (k = 0.83, RL = 58.708 Ohm)

Output power (k = 0.83, RL = 58.708 Ohm)

Wat

t(W

)

0

10

20

Wat

t(W

)

0

2

4

6

× 1e-2Time (s)

0.0 0.5 1.0 1.5

Wat

t(W

)

0

200

400

600

800

Input voltage

Sd+

Sd-

Output voltage

Primary resonant current

Secondary resonant current

-100

0

100

0.0

0.5

1.0

0.0

0.5

1.0

-200

0

200

-10

0

10

× 1e-2Time (s)

0.0 0.5 1.0 1.5-4-2024

Input voltage

Sd+

Sd-

Output voltage

Primary resonant current

Secondary resonant current

-100

0

100

0.0

0.5

1.0

0.0

0.5

1.0

-200

0

200

-10

0

10

× 1e-3Time (s)

4.2 4.4 4.6 4.8 5.0-4-2024

vb(t)

vin(t)vc(t)

va(t)vb(t)

vin(t)vc(t)

va(t)

Fig. 5: Simulation results of input voltages and primary resonant current using parameters given inTable I for k = 0.55 under a time range of 20 ms. The right figures are the zoomed version of the left onein a time range of 4.2−5.2 ms.

Fig. 6: Coils power loss and output power plots within a time range of 20 ms using parameters given inTable I for k = 0.55 and k = 0.83.

k and RL. When the value of k is low, consumed powers increase and vice versa. The k value depends onthe coil distance, therefore components’ rating can limit a flexible coil placement. Power loss ratio is alsonot constant but depends on k. So the device needs a good cooling system to operate at a low couplingfactor. Another point to mention is, the converter creates a low Vs for equal input phases’ amplitude of100 V. This behavior significantly reduces the transfered power to the load RL.

ConclusionA load selection to minimize coil power loss in a variable frequency direct AC/AC resonant converterhas been presented in this paper. The simulation results confirm the validity of analytical calculations.The proper load selection as well as system gain will be used to formulate current reference to controlprimary current amplitude in a closed loop configuration, which leads to a controlled output power. Sincethe load calculation is based on an iterative method, a precalculated lookup table that maps k to Ropt canbe used in the controller. The modulation strategy should also be improved since it produces a low outputpower. A prototype of the converter is currently being manufactured. The output power will be increasedgradually until it reaches 1.7 kW by increasing the input three-phase amplitude. Practical analyses willbe reported in future publications.

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